The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

3.8 Reflection principle

Some of this material is in the chapter of [Kunen] called “Easy consistency proofs”.

Let $\phi (x_1, \ldots , x_ n)$ be a formula of set theory. Let us use the convention that this notation implies that all the free variables in $\phi $ occur among $x_1, \ldots , x_ n$. Let $M$ be a set. The formula $\phi ^ M(x_1, \ldots , x_ n)$ is the formula obtained from $\phi (x_1, \ldots , x_ n)$ by replacing all the $\forall x$ and $\exists x$ by $\forall x\in M$ and $\exists x\in M$, respectively. So the formula $\phi (x_1, x_2) = \exists x (x\in x_1 \wedge x\in x_2)$ is turned into $\phi ^ M(x_1, x_2) = \exists x \in M (x\in x_1 \wedge x\in x_2)$. The formula $\phi ^ M$ is called the relativization of $\phi $ to $M$.

Theorem 3.8.1. Suppose given $\phi _1(x_1, \ldots , x_ n), \ldots , \phi _ m(x_1, \ldots , x_ n)$ a finite collection of formulas of set theory. Let $M_0$ be a set. There exists a set $M$ such that $M_0 \subset M$ and $\forall x_1, \ldots , x_ n \in M$, we have

\[ \forall i = 1, \ldots , m, \ \phi _ i^{M}(x_1, \ldots , x_ n) \Leftrightarrow \forall i = 1, \ldots , m, \ \phi _ i(x_1, \ldots , x_ n). \]

In fact we may take $M = V_\alpha $ for some limit ordinal $\alpha $.

Proof. See [Theorem 12.14, Jech] or [Theorem 7.4, Kunen]. $\square$

We view this theorem as saying the following: Given any $x_1, \ldots , x_ n \in M$ the formulas hold with the bound variables ranging through all sets if and only if they hold for the bound variables ranging through elements of $V_\alpha $. This theorem is a meta-theorem because it deals with the formulas of set theory directly. It actually says that given the finite list of formulas $\phi _1, \ldots , \phi _ m$ with at most free variables $x_1, \ldots , x_ n$ the sentence

\[ \begin{matrix} \forall M_0\ \exists M, \ M_0 \subset M\ \forall x_1, \ldots , x_ n \in M \\ \phi _1(x_1, \ldots , x_ n) \wedge \ldots \wedge \phi _ m(x_1, \ldots , x_ n) \leftrightarrow \phi _1^ M(x_1, \ldots , x_ n) \wedge \ldots \wedge \phi _ m^ M(x_1, \ldots , x_ n) \end{matrix} \]

is provable in ZFC. In other words, whenever we actually write down a finite list of formulas $\phi _ i$, we get a theorem.

It is somewhat hard to use this theorem in “ordinary mathematics” since the meaning of the formulas $\phi _ i^ M(x_1, \ldots , x_ n)$ is not so clear! Instead, we will use the idea of the proof of the reflection principle to prove the existence results we need directly.


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